Grazing-induced crises in hybrid dynamical systems

Tanaka, Gouhei; Tsuji, Shigeki; Aihara, Kazuyuki

doi:10.1016/j.physleta.2009.06.060

Cited by 6 publications

(5 citation statements)

References 19 publications

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“…It is notable that a grazing bifurcation of a limit cycle and a chaotic attractor, which cannot be observed in ordinary smooth dynamical systems, plays a crucial role in the phase transition (Tanaka et al 2009).…”

Section: Introductionmentioning

confidence: 99%

“…Phase transitions between the regimes of relapse and relapse prevention in the IAS therapy model were revealed from a mathematical viewpoint by numerical bifurcation analyses (Tanaka et al 2008). It is notable that a grazing bifurcation of a limit cycle and a chaotic attractor, which cannot be observed in ordinary smooth dynamical systems, plays a crucial role in the phase transition (Tanaka et al 2009). By modifying the IAS therapy model, a competition between AD and AI cells in androgen-poor conditions was considered as a candidate mechanism to enhance the possibility of relapse prevention (Shimada & Aihara 2008).…”

Section: Introductionmentioning

confidence: 99%

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Mathematical modelling of prostate cancer growth and its application to hormone therapy

Tanaka

Hirata

Goldenberg³

et al. 2010

Phil. Trans. R. Soc. A.

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Hormone therapy in the form of androgen deprivation is a major treatment for advanced prostate cancer. However, if such therapy is overly prolonged, tumour cells may become resistant to this treatment and result in recurrent fatal disease. Long-term hormone deprivation also is associated with side effects poorly tolerated by patients. In contrast, intermittent hormone therapy with alternating on-and off-treatment periods is a possible clinical strategy to delay progression to hormone-refractory disease with the advantage of reduced side effects during the off-treatment periods. In this paper, we first overview previous studies on mathematical modelling of prostate tumour growth under intermittent hormone therapy. The model is categorized into a hybrid dynamical system because switching between on-treatment and off-treatment intervals is treated in addition to continuous dynamics of tumour growth. Next, we present an extended model of stochastic differential equations and examine how well the model is able to capture the characteristics of authentic serum prostate-specific antigen (PSA) data. We also highlight recent advances in time-series analysis and prediction of changes in serum PSA concentrations. Finally, we discuss practical issues to be considered towards establishment of mathematical model-based tailor-made medicine, which defines how to realize personalized hormone therapy for individual patients based on monitored serum PSA levels.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mathematical modelling of prostate cancer growth and its application to hormone therapy

Tanaka

Hirata

Goldenberg³

et al. 2010

Phil. Trans. R. Soc. A.

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“…The three-ball contact motion occurs at ¼ 0:5655 mm, the grazing phenomenon occurs and the frequency-response curve undergoes a major change. The 5,7,9,11,13 and with the decreasing bearing clearance. As the bearing maintains in tightening interference fitting conditions, the frequency-response curve near the natural frequency alters its shape significantly (Figure 7(p) and (q)).…”

Section: Resultsmentioning

confidence: 97%

“…The map is discontinuous in the neighborhood of a grazing orbit and has a square root slope singularity on one side of the discontinuity. Tanaka et al 9 demonstrated that a grazing phenomenon of a chaotic attractor can disappear suddenly in dynamical systems including both continuous and discrete components, in which the particular types of sliding and grazing bifurcations of periodic-impact motions are exhibited under parameter variations.…”

Section: Introductionmentioning

confidence: 99%

Grazing contact researches for a spindle-ball bearing system

Gao

Nong

Meng

et al. 2013

Proceedings of the Institution of Mechanical Engineers, Part K:

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This article combines the three-order harmonic balance method and eight-order Taylor expansion technique to give the qualitative analysis of responses for the spindle-ball bearing system. The effects of the bearing clearance and initial position of rotor on the frequency-response curves are investigated. As the number of contact balls alters due to the combining actions of the bearing clearance and initial position of rotor, the grazing phenomenon occurs and the frequency-response curve undergoes a major change. The system's characteristic coefficients are also found to sharply grow at the point the number of contact balls alters. They therefore can be treated as the signs to indicate the occurrence of the grazing phenomenon, which is helpful for the researchers to study the non-linear and non-smooth properties of the spindle-ball bearing system.

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“…10 We have also clarified that the mechanism of the transition is related to a grazing bifurcation, which is unique to hybrid systems. 11 As shown in Figure 4 The partial differential equations model The initial model of IAS, as shown in Equations (1)-(6), can be extended to a partial differential equations model that takes into consideration the spatial distribution of tumour cells in prostate cancer, 12 where a tumour is modelled as a radially symmetric sphere densely packed with AD and AI cells. The model equations can be written as follows:…”

Section: Mathematical Models For Iasmentioning

confidence: 99%

Mathematically modelling and controlling prostate cancer under intermittent hormone therapy

Hirata¹,

Tanaka²,

Bruchovsky³

et al. 2012

Asian J Androl

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In this review, we summarize our recently developed mathematical models that predict the effects of intermittent androgen suppression therapy on prostate cancer (PCa). Although hormone therapy for PCa shows remarkable results at the beginning of treatment, cancer cells frequently acquire the ability to grow without androgens during long-term therapy, resulting in an eventual relapse. To circumvent hormone resistance, intermittent androgen suppression was investigated as an alternative treatment option. However, at the present time, it is not possible to select an optimal schedule of on- and off-treatment cycles for any given patient. In addition, clinical trials have revealed that intermittent androgen suppression is effective for some patients but not for others. To resolve these two problems, we have developed mathematical models for PCa under intermittent androgen suppression. The mathematical models not only explain the mechanisms of intermittent androgen suppression but also provide an optimal treatment schedule for the on- and off-treatment periods.

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Grazing-induced crises in hybrid dynamical systems

Cited by 6 publications

References 19 publications

Mathematical modelling of prostate cancer growth and its application to hormone therapy

Mathematical modelling of prostate cancer growth and its application to hormone therapy

Grazing contact researches for a spindle-ball bearing system

Mathematically modelling and controlling prostate cancer under intermittent hormone therapy

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